Beurling moving averages and approximate homomorphisms

Abstract

Abstract. The theory of regular variation, in its Karamata and Bojanić-Karamata/de Haan forms, is long established and makes essential use of homomorphisms. Both forms are subsumed within the recent theory of Beurling regular variation, developed further here, especially certain moving averages occurring there. Extensive use of group structures leads to an algebraicization not previously encountered here, and to the approximate homomorphisms of the title. Dichotomy results are obtained: things are either very nice or very nasty. Quanti…er weakening is extended, and the degradation resulting from working with limsup and liminf, rather than assuming limits exist, is studied.

Beurling slow and regular variation

“…We may now state Beurling's extension to Wiener's Tauberian theorem (for background and further results, see § 10.1 and [19,45]).…”

“…Such moving averages are Riesz (typical) means, and here ϕ increasing to ∞ is natural in context. For a textbook account, see [30] and the recent [19]; for applications, in analysis and probability theory, see [8,20]. The prototypical case is ϕ(x) = x α (0 < α < 1); this corresponds to X ∈ L 1/α for the probability law of X.…”

“…Rearranging yields Taking d(x) = e c(x) , we obtain the desired representation. To check this, without loss of generality we now take d(x) = 1, and continue by substituting u = x + sϕ(x) to obtain, from (18) and (19) The proof above remains valid when ρ = 0 for arbitrary ϕ ∈ SN, irrespective of whether ϕ is bounded away from zero or not. Since ϕ is itself ϕ-regularly varying with corresponding index ρ = 0, we have an alternative to the Bloom-Shea representation of ϕ via the de Bruijn-Karamata representation.…”

“…For further background and more references, see our arXiv papers [17,18] and [19]. Despite the numerous positive results below, the question that motivated this study remains open, at least in its original form.…”

General regular variation, Popa groups and quantifier weakening

“…We recall the definition of Beurling slowly varying functions ϕ (see e.g. [BinGT § 2.11], [BinO7]): these are positive, measurable or Baire (i.e. have the Baire property, BP), are o(x) at infinity (or O(x), depending on context), and, with x • ϕ t := x + tϕ(x)…”